Efficient quantile regression with auxiliary information

We discuss efficient estimation in quantile regression models where the quantile regression function is modeled parametrically. Additionally we assume that auxiliary information is available in the form of a conditional constraint. This is, for example, the case if the mean regression function or the variance function can be modeled parametrically, e.g. by a line or a polynomial. In this paper we describe efficient estimators of parameters of the quantile regression function for general conditional constraints and for examples of more specific constraints. We do this more generally for a model with responses missing at random, for which an efficient estimator is provided by a complete case statistic. This covers the usual model as a special case. We discuss several examples and illustrate the results with simulations.